The orbit classification conjecture for horospherical varieties

Let $X$ be an affine or projective irreducible normal horospherical variety of complexity zero, with an action of a reductive group $G$. For a $G$-orbit $\mathcal{O}$, let $\Gamma_G(\mathcal{O})$ be the submonoid of the modified divisor class group $\mathrm{Cl}_G(X)$ generated by the classes of the prime $G$-invariant divisors that do not contain a point of $\mathcal{O}$.

The orbit classification conjecture. Two $G$-orbits $\mathcal{O}$ and $\mathcal{O}'$ lie in the same $\mathrm{Aut}(X)^0$-orbit if and only if

$$\Gamma_G(\mathcal{O})=\Gamma_G(\mathcal{O}')$$

\nand the dimensions of the tangent spaces at points of these orbits are equal.

The conjecture strengthens the necessary condition proved in the paper by adding equality of tangent-space dimensions, which is needed because equality of the monoids alone does not distinguish all automorphism-group orbits, as shown by an explicit affine example. It is compared with a conjecture of Bien and Gontsov–Shafarevich; its general validity remains open.